Power Series - Representation of Functions - Calculus 2

This calculus 2 video tutorial provides a basic introduction into the representation of functions as power series. It explains how to represent a function as a power series centered at 0 and centered at some value c. It also explains how to approximate function values using geometric power series. This video also explains how to find the radius of convergence and the interval of convergence of a geometric power series. To write a power series from a function, you need to write the function in the form of the formula for the infinite sum of a geometric series. To find the interval of convergence, find the common ratio represented by some variable expression of x and set it less than 1 and solve the inequality. This video also includes examples and practice problems of adding and subtracting power series as well as using partial fraction decomposition.
Series Tests - Practice Problems:
• Calculus 2 - Geometric...
Taylor & Maclaurin Polynomials:
• Taylor Polynomials & M...
Taylor's Remainder Theorem:
• Taylor's Remainder The...
Power Series - Interval Convergence:
• Power Series - Finding...
Power Series - Derivatives & Integrals:
• Power Series - Differe...
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Power Series - Function Representation:
• Power Series - Represe...
Finding The Power Series:
• Finding Power Series B...
Power Series - Representation By Integration:
• Power Series Represent...
Power Series - Representation By Natural Logs:
• Power Series Represent...
Taylor & Maclaurin Series:
• Taylor Series and Macl...
Binomial Series:
• Binomial Series
____________________________________
Parametric Equations:
• Parametric Equations I...
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[@TheOrganicChemistryTutor]

Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections Next Video: kzbin.info/www/bejne/hp6rkoGgeaeblcU

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[@eggxecution]

done thanks! I realized that you can add constants to make denominator into 1-x but it kind of changes the overall power series function, that's why I think *the unwritten rule when transforming a function into a power series function is that when transforming them into a/1-r you should not change the overall value of the function*, therefore it's preferred to just multiply or divide num/den but when adding a constant, make sure it is =0, just like 1-1 =0, because adding a higher value to make dennominator 1-x introduces a new constant and changes the resulting power series. I got a different answer on my side.

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Professor Organic Chemistry Tutor, thank you for an awesome introduction and analysis of the Power Series Representation of Functions in Calculus Two. Power Series Representation of Functions has played an important role in the development of Calculus. Once you define a function with a Power Series, it is also important to analyze the characteristics of the function. Continuity, Differentiability and integral ability are important characteristics of a function in Calculus. This is an error free video/lecture on KZbin TV with the Organic Chemistry Tutor.

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Thank you for the wonderful explanation as always but I have a question. What if the intervals are given for example -1

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[@abrarmahi]

if you were confused like I was (maybe I'm just stupid) . @ 5:04 r (lowercase) is the rate of the geometric sequence, and R (capital) is the radius of convergence.

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[@philipwest2551]

A quick question: When we write Sum (0, infinity) x^n = 1/(1-x) do we include the value x=0 or not?

[@Damian-vy8it]

im not 100% sure but I think there is a mistake 20:00, when u factor 1/3 from the denomnator that should result in a 3/(1 - (x/3)) which the should turn into 3(x/3)^n which is x^n / 3^(n -1)

[@sedonadettwiler8127]

Clarification Question: at about 43:30 you say that r is -x/2. does the x^3 in the numerator of the original function not impact what r is? is r the same for both x^3/x+2 and 1/x+2?

[@hamdymubarak3082]

Are all power series representing a function have an open interval or do we have to test the endpoints in every problem?

[@pythonlearner2098]

In the last example can not I just use completing square in the denominator and solve it as the previous examples ?

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His voice is nostalgic

[@arekkrolak6320]

this is all good material, but now how do you deal with functions like sin(x)? :)

[@kithenry]

for the number at 21:27 , some one tell me why we dont use (1+2) as a replacement for three to make the function fit the GP template (a/(1-r))

[@minhluanne7462]

because we need to ensure that C value is=0 following the problem's request

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차근차근 잘가르쳐준다

[@yona9257]

on the last example can someone explain why he wrote one over x plus two instead of -1 over x plus two

[@be4913]

watching this the night before my final lol

[@CarmensProjects]

What if the function you're given is xln(1+x) and the center is at 2? Sorry if this is a dumb question, this is during the COVID-19 quarantine and my calc professor gave up on us. No lectures, just homework that doesn't explain anything.

[@user-hw7rg4xu6o]

At 48:42, why it shows as 1/x+2 instead of -1/x+2 somehow?

[@mateoortega719]

it's brought back at the end of the video

[@MTNtf2]

So is c 0 for the last example? I dont understand how we choose c.

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does anyone know if you can turn an iterative function into a function? (I'm trying to turn x=x+195y into a function.)

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[@duckmancow1235]

cool

[@yona9257]

for example number three why is it only x^3 and not -x^3 for R

[@thedarkdev3439]

The formula is a / 1 - r; the negative in the " 1 - r " is part of it, so you would not include it. However, if it was, for example, a / 1- ( -r ) , you would include it as such.

[@Thefakisgoinon]

I have a function f(x)=1/(2-x) and i rewrite it as f(x)=1/(1-(x-1) which gives us a=1 and r=(x-1). However in my textbook it says that a=1/2 and r=(x/2)?

[@samuelgunter]

do people use calculus in organic chemistry?

[@aidilshahfiq2950]

what is different when function 1/(1-x) and 1/((1-x)^2) when convert to power series?

[@Gwyn94]

1/((1-x)^2) is the derivative of 1/(1-x). To find the power series of 1/((1-x)^2) you need to differentiate the power series of 1/(1-x). So it would be n(x^(n-1)), and then you change your summation start point from n=0 to n=1. Always increase your start point by 1 when you differentiate a power series